I just had the same issue. (But solved it read the last paragraph)
Linux Mint 20
CPU: 2920X
GPU: R9 390
EVE installed on Lutris using DXVK
I launched the launcher it downloaded about 1GB of updates (i havent loged in in about a week)
launched the client
wached a black screen with a rotating cycle in the bottom left for 5min
got to the carector select and selected carrector
waited 5 more minutes
got into game.
after trying DX9, different toons. I then closed and reopened the launcher and it downloaded a 500GB pach.
So if you are having trouble on linux (normally after Down Time) close and reopen your launcher.